Angle between curve and sphere

[autre]

I need to find the angle between a curve x^2 + y^2 =1, z = sqrt(2)x/y and a sphere z^2 + y^2 + z^2 = 1. I found two points of intersection, (0,1,0) and (0,-1,0), so at each of these points I assume that there's a tangent line to the curve and a tangent plane to the sphere. Then I'm guessing I'd find the angle between the direction vector of the tangent line to the curve and the normal vector of the tangent plane. Am I on the right track?

I have the direction vector of the tangent line to the curve at (0,1,0) as <1, 0, sqrt(2)> and at (0,-1,0) as <-1,0,-sqrt(2)>. I'm not sure where to go from here.

 

[micromass]

I need to find the angle between a curve x^2 + y^2 =1, z = sqrt(2)x/y and a sphere z^2 + y^2 + z^2 = 1. I found two points of intersection, (0,1,0) and (0,-1,0), so at each of these points I assume that there's a tangent line to the curve and a tangent plane to the sphere. Then I'm guessing I'd find the angle between the direction vector of the tangent line to the curve and the normal vector of the tangent plane. Am I on the right track?

Yes, that's good.

I have the direction vector of the tangent line to the curve at (0,1,0) as <1, 0, sqrt(2)> and at (0,-1,0) as <-1,0,-sqrt(2)>. I'm not sure where to go from here.

Well, find the tangent plane to the sphere at the points (0,1,0) and (0,-1,0).

 

[autre]

Oh, so that'll just be a 90 degree angle in both cases, since <+-1, 0, +-sqrt(2)>.<0,+-2,0> = 0 so arccos0 = pi/2.